Beatles' Apple Corps was seeking financial damages and court orders to stop Apple Computer using the "apple" marks in connection with the website that allows iPod users to download and save thousands of songs.
Apple Corps -- owned by former Beatles Paul McCartney, Ringo Starr and the widows of bandmates John Lennon and George Harrison -- had accused the US company of breaching a trademark agreement between the firms by promoting music products on the Internet.
The verdict could cost Apple Corps more than three million pounds (5.
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6 million dollars, 4.4 million euros) once all bills have been finalised.
On Monday, it agreed to pay Apple Computer's costs, so far estimated at about two million pounds.
Apple Computer is the market leader for music downloads, with around three million songs being downloaded each day from its iTunes service.
Judge Edward Mann, a self-confessed iPod owner, granted The Beatles' firm permission to appeal against his ruling.
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